One-way MANOVA | SAS Data Analysis Examples

Version info: Code for this page was tested in SAS 9.3

MANOVA is used to model two or more dependent variables that are
continuous with one or more categorical predictor variables.

Please note: The purpose of this page is to show how to use various data
analysis commands. It does not cover all aspects of the research process which
researchers are expected to do. In particular, it does not cover data
cleaning and checking, verification of assumptions, model diagnostics or
potential follow-up analyses.

Examples of one-way multivariate analysis of variance

Example 1. A researcher randomly assigns 33 subjects to one of three groups. The first group
receives technical dietary information interactively from an on-line website. Group
2 receives the same information from a nurse practitioner, while group 3 receives the
information from a video tape made by the same nurse practitioner. The
researcher looks
at three different ratings of the presentation, difficulty, usefulness and importance, to determine
if there is a difference in the modes of presentation. In particular, the researcher is
interested in whether the interactive website is superior because that is the most cost-effective
way of delivering the information.

Example 2. A clinical psychologist recruits 100 people who suffer from
panic disorder into his study. Each subject receives one of four types of
treatment for eight weeks. At the end of treatment, each subject
participates in a structured interview, during which the clinical psychologist
makes three ratings: physiological, emotional and cognitive. The
clinical psychologist wants to know which type of treatment most reduces the
symptoms of the panic disorder as measured on the physiological, emotional and
cognitive scales. (This example was adapted from Grimm and Yarnold, 1995,
page 246.)

Description of the data

Let’s pursue Example 1 from above.

We have a data file, manova,
with 33 observations on three response variables.
The response variables are ratings of useful, difficulty and importance.
Level 1 of the group variable is the treatment group, level 2 is control group 1 and
level 3 is control group 2.

Let’s look at the data. It is always a good idea to start with descriptive
statistics.

Analysis methods you might consider

Below is a list of some analysis methods you may have
encountered. Some of the methods listed are quite reasonable, while others
have either fallen out of favor or have limitations.

MANOVA – This is a good option if there are two or more continuous
dependent variables and one categorical predictor variable.

Discriminant function analysis – This is a reasonable option and is equivalent to
a one-way MANOVA.

The data could be reshaped into long format and analyzed as a multilevel
model.

Separate univariate ANOVAs – You could analyze these data using separate univariate
ANOVAs for each response variable. The univariate ANOVA
will not produce multivariate results utilizing information from all variables
simultaneously. In addition, separate univariate tests are generally less powerful
because they do not take into account the inter-correlation of the dependent
variables.

One-way MANOVA

We will use proc glm to run the one-way MANOVA. We will list the
variable group on the class statement to indicate that it is a
categorical predictor variable. We use the ss3 option on the
model statement to get only the Type III sums of squares in the output.
We use some contrast statements to specify two contrasts in which we are
interested. We will discuss these when we see their output. We use
the first manova statement to obtain all of the multivariate tests that
SAS offers; we use the second manova statement to run the multivariate
tests using only the variables useful and importance.

Because the output is very long, we will break it up and discuss the
different sections individually. Please also see our
Annotated Output: SAS MANOVA.

The above output shows the three one-way ANOVAs. While none of the three
ANOVAs were statistically significant at the alpha = .05 level,
in particular, the F-value for difficulty was less than 1.

We also see the results of the two contrast statements. The first contrast
compares the treatment group (group 1) to the average of the two control groups
(groups 2 and 3). The second contrast compares the two control groups.
The first contrast is statistically significant for useful and importance, but not for
difficulty. The second contrast is not
statistically significant for any of the dependent variables.

The overall multivariate test is significant, which means that differences
between the levels of the variable group exist. To find where the
differences lie, we will follow up with several post-hoc tests. We will begin with the multivariate test of group 1 versus the
average of groups 2 and 3.

The multivariate test with useful and importance as dependent
variables and group as the independent variable is statistically
significant.

We can use the lsmeans statement to obtain adjusted predicted values
for each of the dependent variables for each of the groups. These values can be
helpful in seeing where differences between levels of the predictor variable are
and describing the model.

In each of the three columns above, we see that the predicted means for
groups 2 and 3 are very similar; the predicted mean for group 1 is higher than
those for groups 2 and 3.

In the examples below, we obtain the differences in the means for each of the
dependent variables for each of the control groups (groups 2 and 3) compared to
the treatment group (group1), by specifying group 1 to be the reference group
(called “control” by SAS, confusingly for this scenario). With respect to the dependent variable useful,
the difference between the means for control group 1 versus the treatment group
is approximately -2.59 (15.53 – 18.12). The difference between the means for
control group 2 versus the treatment group is approximately -2.77 (15.35 –
18.12). With respect to the dependent variable difficulty, the
difference between the means for control group 1 versus the treatment group is
approximately -0.61 (5.58 – 6.19). The difference between the means for control
group 2 versus the treatment group is approximately -0.82 (5.37 – 6.19).

None of the three ANOVAs were statistically significant at the alpha = .05 level.
In particular, the F-ratio for difficulty was less than 1.

Things to consider

One of the assumptions of MANOVA is that the response variables come
from group populations that are multivariate normal distributed.
This means that each of the dependent variables is normally distributed within
group, that any linear combination of the dependent variables is normally
distributed, and that all subsets of the variables must be multivariate
normal. With
respect to Type I error rate, MANOVA tends to be robust to minor
violations of the multivariate normality assumption.

The homogeneity of population covariance matrices (a.k.a. sphericity) is another assumption.
This implies that the population variances and covariances of all dependent
variables must be equal in all groups formed by the independent variables.

Small samples can have low power, but if the multivariate normality assumption is met,
the MANOVA is generally more powerful than separate univariate tests.

There are at least five types of follow-up analyses that can be done
after a statistically significant MANOVA. These include multiple
univariate ANOVAs, stepdown analysis, discriminant analysis, dependent
variable contribution, and multivariate contrasts.